MEDIUM

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Which type of conic section is formed when a plane cuts the nappe of the cone and makes an angle $θ$ with nappe where $α < θ < 90 °$ ? Determine the eccentricity of $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ .

Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The focus of the parabola

{(y + 1)}^{2} = - 8 (x + 2)

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

P Q

be a double ordinate of the parabola,

y^{2} = - 4 x,

where

P

lies in the second quadrant. If

R

divides

P Q

in the ratio

2 : 1,

then the locus of

R

is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The focus of the parabola

y^{2} - 4 y - x + 3 = 0

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

P

and

Q

are two distinct points on the parabola,

y^{2} = 4 x

, with parameters

t

and

t_{1}

, respectively. If the normal at

P

passes through

Q

, then the minimum value of

t_{1}^{2}

, is

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

A cable of a suspension bridge is in the form of a parabola whose span is $40$ metres. The road way is $5$ metres below the lowest point of the cable. An extra support is provided across the cable $30$ metres above the ground level. Find the length of the support if the height of the pillars are $55$ metres.

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

A chord is drawn through the focus of the parabola

y^{2} = 6 x

such that its distance from the vertex of this parabola is

\frac{\sqrt{5}}{2}

, then its slope can be

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

Suppose the parabola

{(y - k)}^{2} = 4 (x - h),

with vertex

A

, passes through

O = (0, 0)

and

L = (0, 2) .

Let

D

be an end point of the latus rectum. Let the

y

-axis intersect the axis of the parabola at

P

. Then

∠ P D A

is equal to

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The vertex of the parabola

y = x^{2} - 2 x + 4

is shifted

p

units to the right and then

q

units up. If the resulting point is

(4, 5),

then the values of

p

and

q

respectively are

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

Let

O

be the vertex and

Q

be any point on the parabola,

x^{2} = 8 y

. If the point

P

divides the line segment

O Q

internally in the ratio

1 : 3

, then the locus of

P

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

Let

P (4, - 4)

and

Q (9, 6)

be two points on the parabola,

y^{2} = 4 x

and let

X

be any point on the arc

P O Q

of this parabola, where

O

is the vertex of this parabola, such that the area of

∆ P X Q

is maximum. Then this maximum area (in sq. units) is :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The centres of those circles which touch the circle,

x^{2} + y^{2} - 8 x - 8 y - 4 = 0,

externally and also touch the

x

- axis, lie on

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

a x^{2} + 2 h x y + b y^{2} - 82 x + 98 y + 144 = 0

is the equation of a parabola with focus

(2, - 3)

and directrix

3 x - 2 y + 5 = 0,

then

a x^{2} + 2 h x y + b y^{2} = 0

represents

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

If one end of a focal chord of the parabola,

y^{2} = 16 x

is at

(1, 4)

, then the length of this focal chord is

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

y = m x + c

is the normal at a point on the parabola

y^{2} = 8 x

whose focal distance is

8

units, then

|c|

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The equation of the directrix of the parabola

x^{2} - 4 x - 3 y + 10 = 0

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The area (in sq. units) of an equilateral triangle inscribed in the parabola

y^{2} = 8 x,

with one of its vertices on the vertex of this parabola is

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The vertex of the parabola

y = (x - 2) (x - 8) + 7

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

Let

A (4, - 4)

and

B (9, 6)

be points on the parabola,

y^{2} = 4 x .

Let

C

be chosen on the arc

A O B

of the parabola, where

O

is the origin, such that the area of

Δ A C B

is maximum. Then, the area (in sq. units) of

Δ A C B

, is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The focus of the parabola

y = 2 x^{2} + x

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Conic Section

The equation

a x^{2} + 4 x y + y^{2} + a x + 3 y + 2 = 0

represents a parabola if

a

Which type of conic section is formed when a plane cuts the nappe of the cone and makes an angle θ with nappe where α<θ<90°? Determine the eccentricity of x24+y29=1.

Important Questions on Parabola

Learn and Prepare for any exam you want !

Which type of conic section is formed when a plane cuts the nappe of the cone and makes an angle $θ$ with nappe where $α < θ < 90 °$ ? Determine the eccentricity of $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$ .